How To Add Multiple Fractions With Different Denominators

we're asked to subtract,simplify the answer and write as a mixed number. so we have two mixed numbershere, and like when you're adding mixed numbers, youcan do it two ways. you can either turn both ofthese into improper fractions

How To Add Multiple Fractions With Different Denominators, and then subtract, or you cansubtract the whole number part and then the fraction part,so let's do it that way. so just to remind ourselves,5 and 5/8 is the exact same thing-- let me write it herein yellow-- as 5 plus 5/8.

these are the exactsame thing. and subtracting-- let me writethis subtraction sign in blue-- and then we'resubtracting 2 and 1/5, but 2 and 1/5 is the exact samething as 2 plus 1/5. that's what 2 and1/5 really is. now if we're subtracting, thatmeans we have to subtract 2 and we're subtracting the 1/5. distributive property. we're subtractingboth of them.

so we get 5 plus 5/8, and thenwe distribute the negative sign, minus 2 minus 1/5. and now we can rearrange it. we can say this is the samething as-- let me do it in orange-- as 5 minus 2, that'sthe whole number parts, and then you have plus5/8 minus 1/5. now, 5 minus 2, that's prettystraightforward. that's just 3. but what's 5/8 minus 1/5?

let's do that over here. so we have 5/8 minus 1/5. now, just like when you'readding fractions, when you're subtracting fractions, youhave to have a common denominator. 8 and 5 are not the samedenominator so we have to find the least common multipleof these two numbers. if we do a prime factorizationof 8, you get 2 times 4. 4 is 2 times 2.

so 8 is 2 times 2 times 2. that's if you break it down intoits constituent primes. 5 is just 5, because itis a prime number. so the least common multiple of8 and 5 has to have a 5 and three 2's in it. or another way of saying it, ithas to be 5 times 2 times 2 times 2, which is the samething as 5 times 8. so the least common multiple isgoing to be-- because these guys share no common factorsother than 1 really.

the least common multiple isjust going to be 8 times 5. so the least common multipleis going to be 40. it's going to be 40. now, to go from 8 to 40, youhave to multiply by 5. then we have to do the samething to the numerator. if we multiplied 8 times 5,we have to multiply 5 times 5 to get 25. so in either case, we'remultiplying times 5, the numerator and the denominator.

now, to go from 5 to 40, we haveto multiply by 8, so we also have to do that for thenumerator, so 1 times 8 is 8. so now we have 25/40 minus40/40, which would be equal to-- we're going to have40 as the denominator. 25 minus 8 is 17. so 5/8 minus 1/5, which is thesame thing as 25/40 minus 8/40, is equal to 17/40. and 17 is a prime number so it'snot going to share any common factors with 40, or atleast 40 isn't divisible by 17

either, so this is as simplifiedas we can get. so this part, the fraction part,is 3 and the fraction part is 17/40. and we're done. and this worked. we were able to separate thisout and do it pretty straightforward, because in thissituation, the fraction part that we're subtracting fromthe other fraction part was smaller.

if it was larger, it would'vegotten a little bit messy. maybe we would have ended upwith a negative number and we would've done all this otherstuff, so that's why sometimes it's easier to go straightto an improper fraction for both of them. but in this case, it workedout doing it this way.